Theorem ltaprg 7734

Description: Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by Jim Kingdon, 26-Dec-2019.)

Assertion

Ref	Expression
ltaprg	|- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A <P B <-> ( C +P. A ) <P ( C +P. B ) ) )

Proof of Theorem ltaprg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltaprlem 7733	. . 3 |- ( C e. P. -> ( A <P B -> ( C +P. A ) <P ( C +P. B ) ) )
2	1	3ad2ant3 1023	. 2 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A <P B -> ( C +P. A ) <P ( C +P. B ) ) )
3		ltexpri 7728	. . . . 5 |- ( ( C +P. A ) <P ( C +P. B ) -> E. x e. P. ( ( C +P. A ) +P. x ) = ( C +P. B ) )
4	3	adantl 277	. . . 4 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C +P. A ) <P ( C +P. B ) ) -> E. x e. P. ( ( C +P. A ) +P. x ) = ( C +P. B ) )
5		simpl1 1003	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. P. /\ ( ( C +P. A ) +P. x ) = ( C +P. B ) ) ) -> A e. P. )
6		simprl 529	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. P. /\ ( ( C +P. A ) +P. x ) = ( C +P. B ) ) ) -> x e. P. )
7		ltaddpr 7712	. . . . . . 7 |- ( ( A e. P. /\ x e. P. ) -> A <P ( A +P. x ) )
8	5, 6, 7	syl2anc 411	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. P. /\ ( ( C +P. A ) +P. x ) = ( C +P. B ) ) ) -> A <P ( A +P. x ) )
9		addassprg 7694	. . . . . . . . . . . 12 |- ( ( C e. P. /\ A e. P. /\ x e. P. ) -> ( ( C +P. A ) +P. x ) = ( C +P. ( A +P. x ) ) )
10	9	3com12 1210	. . . . . . . . . . 11 |- ( ( A e. P. /\ C e. P. /\ x e. P. ) -> ( ( C +P. A ) +P. x ) = ( C +P. ( A +P. x ) ) )
11	10	3expa 1206	. . . . . . . . . 10 |- ( ( ( A e. P. /\ C e. P. ) /\ x e. P. ) -> ( ( C +P. A ) +P. x ) = ( C +P. ( A +P. x ) ) )
12	11	adantrr 479	. . . . . . . . 9 |- ( ( ( A e. P. /\ C e. P. ) /\ ( x e. P. /\ ( ( C +P. A ) +P. x ) = ( C +P. B ) ) ) -> ( ( C +P. A ) +P. x ) = ( C +P. ( A +P. x ) ) )
13		simprr 531	. . . . . . . . 9 |- ( ( ( A e. P. /\ C e. P. ) /\ ( x e. P. /\ ( ( C +P. A ) +P. x ) = ( C +P. B ) ) ) -> ( ( C +P. A ) +P. x ) = ( C +P. B ) )
14	12, 13	eqtr3d 2240	. . . . . . . 8 |- ( ( ( A e. P. /\ C e. P. ) /\ ( x e. P. /\ ( ( C +P. A ) +P. x ) = ( C +P. B ) ) ) -> ( C +P. ( A +P. x ) ) = ( C +P. B ) )
15	14	3adantl2 1157	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. P. /\ ( ( C +P. A ) +P. x ) = ( C +P. B ) ) ) -> ( C +P. ( A +P. x ) ) = ( C +P. B ) )
16		simpl3 1005	. . . . . . . 8 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. P. /\ ( ( C +P. A ) +P. x ) = ( C +P. B ) ) ) -> C e. P. )
17		addclpr 7652	. . . . . . . . 9 |- ( ( A e. P. /\ x e. P. ) -> ( A +P. x ) e. P. )
18	5, 6, 17	syl2anc 411	. . . . . . . 8 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. P. /\ ( ( C +P. A ) +P. x ) = ( C +P. B ) ) ) -> ( A +P. x ) e. P. )
19		simpl2 1004	. . . . . . . 8 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. P. /\ ( ( C +P. A ) +P. x ) = ( C +P. B ) ) ) -> B e. P. )
20		addcanprg 7731	. . . . . . . 8 |- ( ( C e. P. /\ ( A +P. x ) e. P. /\ B e. P. ) -> ( ( C +P. ( A +P. x ) ) = ( C +P. B ) -> ( A +P. x ) = B ) )
21	16, 18, 19, 20	syl3anc 1250	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. P. /\ ( ( C +P. A ) +P. x ) = ( C +P. B ) ) ) -> ( ( C +P. ( A +P. x ) ) = ( C +P. B ) -> ( A +P. x ) = B ) )
22	15, 21	mpd 13	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. P. /\ ( ( C +P. A ) +P. x ) = ( C +P. B ) ) ) -> ( A +P. x ) = B )
23	8, 22	breqtrd 4071	. . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. P. /\ ( ( C +P. A ) +P. x ) = ( C +P. B ) ) ) -> A <P B )
24	23	adantlr 477	. . . 4 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C +P. A ) <P ( C +P. B ) ) /\ ( x e. P. /\ ( ( C +P. A ) +P. x ) = ( C +P. B ) ) ) -> A <P B )
25	4, 24	rexlimddv 2628	. . 3 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C +P. A ) <P ( C +P. B ) ) -> A <P B )
26	25	ex 115	. 2 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( C +P. A ) <P ( C +P. B ) -> A <P B ) )
27	2, 26	impbid 129	1 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A <P B <-> ( C +P. A ) <P ( C +P. B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2176   E.wrex 2485   class class class wbr 4045  (class class class)co 5946   P.cnp 7406   +P. cpp 7408   <P cltp 7410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-eprel 4337  df-id 4341  df-po 4344  df-iso 4345  df-iord 4414  df-on 4416  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-irdg 6458  df-1o 6504  df-2o 6505  df-oadd 6508  df-omul 6509  df-er 6622  df-ec 6624  df-qs 6628  df-ni 7419  df-pli 7420  df-mi 7421  df-lti 7422  df-plpq 7459  df-mpq 7460  df-enq 7462  df-nqqs 7463  df-plqqs 7464  df-mqqs 7465  df-1nqqs 7466  df-rq 7467  df-ltnqqs 7468  df-enq0 7539  df-nq0 7540  df-0nq0 7541  df-plq0 7542  df-mq0 7543  df-inp 7581  df-iplp 7583  df-iltp 7585

This theorem is referenced by:  prplnqu  7735  addextpr  7736  caucvgprlemcanl  7759  caucvgprprlemnkltj  7804  caucvgprprlemnbj  7808  caucvgprprlemmu  7810  caucvgprprlemloc  7818  caucvgprprlemexbt  7821  caucvgprprlemexb  7822  caucvgprprlemaddq  7823  caucvgprprlem1  7824  caucvgprprlem2  7825  ltsrprg  7862  gt0srpr  7863  lttrsr  7877  ltsosr  7879  ltasrg  7885  prsrlt  7902  ltpsrprg  7918  map2psrprg  7920